"Somewhere inside the digits of pi is a representation for all of us -- the atomic coordinates of all our atoms, our genetic code, all our thoughts, all our memories. Given this fact, all of us are alive, and hopefully happy, in pi. Pi makes us live forever. We all lead virtual lives in pi. We are immortal." - Cliff Pickover
They say that the square root of 2 is not encoded within pi ...but I suspect that it probably is, as a sequence of separated digits
The invariance of perception automatically includes pi.
To say that reality is an illusion, or simulation, implies that there exists a level of reality that is ...NOT an illusion.
The non-simulated reality would thus be at the top of the hyperreality hierarchy, where the buck stops. The reality of illusion must necessarily be a simulation, a representative transformation from the simulated abstract to the illusionary "concrete". Thus the invariance of perception still exists even if reality is an illusion; there necessarily exists at least one non-illusionary existence.